Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

EFX Exists for Three Types of Agents

Vishwa Prakash HV, Pratik Ghosal, Prajakta Nimbhorkar, Nithin Varma

Abstract

We study the problem of finding an envy-free allocation of indivisible goods among agents with additive valuations. We focus on the fairness notion of envy-freeness up to any good (EFX). A central open question in fair division is whether EFX allocations always exist for any number of agents. While EFX has been established for three agents [CGM24] and for any number of agents with at most two distinct valuations [Mah23], its existence in more general settings remains open. In this paper, we make significant progress by proving that EFX allocations exist for any number of agents when there are at most three distinct additive valuations. This result simultaneously generalizes both the three-agent case and the two-type case, settling an open question in the field (see [Mah23]).

EFX Exists for Three Types of Agents

Abstract

We study the problem of finding an envy-free allocation of indivisible goods among agents with additive valuations. We focus on the fairness notion of envy-freeness up to any good (EFX). A central open question in fair division is whether EFX allocations always exist for any number of agents. While EFX has been established for three agents [CGM24] and for any number of agents with at most two distinct valuations [Mah23], its existence in more general settings remains open. In this paper, we make significant progress by proving that EFX allocations exist for any number of agents when there are at most three distinct additive valuations. This result simultaneously generalizes both the three-agent case and the two-type case, settling an open question in the field (see [Mah23]).

Paper Structure

This paper contains 22 sections, 27 theorems, 11 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. EFX Allocations:
  3. Our Contributions
  4. Progress Measure:
  5. Pseudo-Cycles:
  6. Making Progress Through Regress, and Virtual Bundles:
  7. Related Work
  8. Preliminaries
  9. Non-degenerate instances efx_3efx_3_simple
  10. Pareto Improvements:
  11. Three Types of Agents
  12. Progress Measures
  13. Pseudo-Cycles
  14. Elementary Cases for Pareto and $\phi$-Improvements
  15. Three Sources in $E_X$
  16. ...and 7 more sections

Key Result

Theorem 1

An EFX allocation always exists for $n$ agents when there are at most three types of agents, where agents of the same type have identical additive valuations.

Figures (1)

  • Figure 1: Some examples of pseudo-cycles

Theorems & Definitions (58)

  • Theorem 1
  • Definition 1: Bundle, Allocation, and Sub-allocation
  • Definition 2: Pareto domination
  • Definition 3: Envy Graph
  • Definition 4: Strong envy
  • Definition 5: EFX
  • Proposition 1
  • Definition 6: Minimally envied subset efx_3
  • Definition 7: Champion efx_3
  • Definition 8: $g$-champion efx_3
  • ...and 48 more